How do you heal sick survivors?

They will go to the Clinic, the more beds you have the more survivors you can accommodate at the same time.

Improving the Clinic will give it more beds and faster healing times.

There is also the Comprehensive Care Edict which you can active at the Chief's House.

This will heal all sick survivors in the City it costs 150,000 contentment points and there is a 4 hour cooldown before you can use it another time.